Phase transitions in cellular automata : From computer science to statistical physics & back. Nazim Fatès

Loria - BP 239 - 54506 Vandœuvre-lès-Nancy – FRANCE

Computer science

ENS Lyon

[email protected]

Statistical physics

Elementary Cellular Automata (ECA) are discrete dynamical systems.

The percolation phenomenon is a well-studied model that shows phase transitions. In this model, sites are arranged on a diagonal square lattice. A link between two sites can be in two states : open (blocking) or closed (porous).

They are constituted of cells arranged linearly, each cell can be in two states symbolized by 0 and 1. In the infinite case, the cells are located on the bi-infinite line Z. In the finite case considered here, we arrange them in a ring Z/nZ.

In isotropic percolation, the links between two neighbouring sites are closed with probability p and open with probability 1-p. A cluster is a maximal set of connected sites.

The state of the cell is represented by a configuration xt = (xit)∈ Q^{Z/nZ}. Classical ECA: In the classical synchronous regime, at each time step t, each cell i is updated according to a local function f : {0,1}³
{0,1}.

From a given site, what is the probability G(p) to obtain an “infinite cluster” ? There exists a critical probability pc= ½, such that : G(p)=0 for p< pc (dry phase), and G(p)>0 for p> pc (wet phase).

Asynchronous ECA: In the asynchronous regime, at each time step t, each cell i has a probability α to be updated : for all i∈ Z/nZ ,

Directed percolation is an anisotropic variant of percolation where the links between two sites are oriented according to a particular direction. It can be interpreted as a dynamical process where and the xaxis is space and the y-axis represents time.

xit+1 = f( xi-1t, xit, xi+1t )

with probability α

xit+1 = xit

with probability 1-α

The size of the clusters also diverge for a critical probability pc . However, pc[DP] > pc[IP].

E C A 5 0

Illustration of percolation phenomena : courtesy of H. Hinrichsen [2]

α=1

α=0.75

α=0.50 Around criticality, experiments and theory predict that the density d(t) (average number of wet sites) evolves according to p:

E C A 1 8

For the critical phase p= pc , the decrease follows : d(t) ~ t-δ where δ= −0.1595… 0.1595 is known experimentally. For the subcritical phase p< pc, the density vanishes more rapidly d(t) pc, the density stabilizes to an asymptotic value das(p). (2) The asymptotic density das(p) diverges around pc as :

In the experimental work [1], we showed that some ECA are “robust” to the change of synchrony rate α while other ECA displayed a sudden change of behaviour. Seven such ECA were identified ; how to explain their change of dynamics ?

Evolution of ECA 50 with α=60. Time goes from bottom to top.

das(p)~(p- pc )β where β= −0.2765… 0.2765 is known experimentally. The exponents β and δ define a universality class : many different phenomena obeying different microscopic rules exhibit power-laws with the same exponents. What about asynchronous ECA ?

Is the change of behaviour of asynchronous ECA in DP universality class ? Finding the critical synchrony rate αc

Finding the critical exponent β

Discussion openings

&

Computer science Seven ECA that have different behaviours in synchronous mode turn out to have the same behaviour in some parts of their asynchronous regime : asynchronism unveils complexity ! The brutal change of attractor observed when changing α can be explained by the use of techniques from statistical physics. Experiments confirm that there exists a phase transition and that this phase transition is in the directed percolation universality class.

Each curve on this plot is obtained by computing the average of d(t) for ECA 146, on Z=100 runs, with a ring of size N=10 000, for a sampling time T=50 000.

Using the experimental value αc, each point on this plot is obtained by computing the asymptotic density das(α− αc) for ECA 50. The size of the ring, the transient time, the sampling time are adapted for each measurement.

The curve obtained for α=0.675 is almost straight and its slope coincides with the expected slope δ=-0.1595 (dashed line).

The aspect of the curve is linear in a log-log plot and the slope β is “almost” in agreement with the experimentally known value βDP= 0.2765.

Repeating this experiment for the seven ECA, we obtain :

Repeating this experiment for the seven ECA, we also obtain good agreement with directed percolation predictions.

ECA αC

6 18 26 50 0.282 0.713 0.475 0.628

58 0.340

106 0.810

146 0.675

Limits of the protocol Precision of αc is less than 10-3, the increase of precision implies more averaging and longer runs.

Limits of the protocol The values in left part of the curve are overestimated because of the critical slowing down phenomenon : the time needed to reach the asymptotic density grows exponentially as we approach criticality. The right part of the curve shows saturation as the density can not exceed a certain limit.

Linking the two models is still an open problem : how can we relate α and p ? Note that for ECA 6, it is the increase of alpha that leads to the sub-critical phase.

Statistical physics The values of the critical exponents β and δ are only known by means of numerical simulations. Determining if they are rational numbers is still an open problem. This question is, according to P. Grassberger, one of the most challenging problems in the study of phase transitions. Is it possible to determine them from the analytical study of asynchronous CA ?

[1] N. Fatès and M. Morvan, An Experimental Study of Robustness to Asynchronism for Elementary Cellular Automata, Complex Systems, Volume 16, 2005. [2] N.Fatès, Robustesse de la dynamique des systèmes discrets : le cas de l'asynchronisme dans les automates cellulaires, ENS Lyon thesis N° 04ENSL0298, 2004. [3] H. Hinrichsen, Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States, Advances in Physics, Vol. 49, 2000. Thanks to : A.Ballier (ENS Lyon), A. Boumaza (LORIA), W.Bouamama, M. Morvan (ENS Lyon), B. Scherrer (LORIA).